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b(it)h(is)g(a)g(line)f(in)h(latitude-longitude)d(space)k(and)g(has)g(a) f(nonzero)h(con)m(tribution)324 4420 y(to)27 b(the)h(w)m(eigh)m(t)f(in) m(tegrals.)41 b(If)27 b(a)g(grid)f(do)s(es)i(not)f(con)m(tain)g(the)h (p)s(ole)f(explicitly)e(as)j(a)f(grid)324 4541 y(v)m(ertex,)43 b(the)d(p)s(ole)e(con)m(tribution)h(m)m(ust)g(b)s(e)h(added)g(to)f(the) h(appropriate)e(cells.)64 b(The)324 4661 y(p)s(ole)31 b(con)m(tribution)h(can)h(b)s(e)g(computed)f(analytically)-8 b(.)470 4782 y(The)30 b(p)s(ole)f(also)f(creates)i(problems)f(for)g (the)g(searc)m(h)i(and)e(in)m(tersection)g(algorithms)324 4902 y(describ)s(ed)38 b(ab)s(o)m(v)m(e.)59 b(F)-8 b(or)37 b(example,)h(a)f(grid)g(cell)f(that)h(o)m(v)m(erlaps)h(the)g(p)s(ole)f (can)h(result)1894 5251 y(19)p eop %%Page: 20 22 20 21 bop 324 548 a Fq(in)28 b(a)g(noncon)m(v)m(ex)j(cell)d(in)g (latitude-longitude)d(co)s(ordinates.)42 b(The)29 b(cross-pro)s(duct)h (test)324 668 y(describ)s(ed)42 b(ab)s(o)m(v)m(e)g(will)c(fail)h(in)i (this)f(case.)70 b(In)42 b(addition,)f(segmen)m(ts)i(near)e(the)h(p)s (ole)324 789 y(t)m(ypically)30 b(exhibit)h(large)g(c)m(hanges)i(in)e (longitude)g(ev)m(en)i(for)f(v)m(ery)h(short)f(segmen)m(ts.)44 b(In)324 909 y(suc)m(h)31 b(a)f(case,)h(the)g(linear)d (parametrizations)g(used)j(ab)s(o)m(v)m(e)g(result)f(in)f(inaccuracies) h(for)324 1029 y(determining)h(the)i(correct)g(in)m(tersections.)470 1150 y(T)-8 b(o)35 b(a)m(v)m(oid)g(these)i(problems,)e(a)g(co)s (ordinate)g(transformation)e(can)i(b)s(e)h(used)g(p)s(ole-)324 1270 y(w)m(ard)e(of)g(a)f(giv)m(en)h(threshold)g(latitude)f(\(t)m (ypically)f(within)h(one)h(degree)h(of)e(the)h(p)s(ole\).)324 1391 y(A)e(p)s(ossible)g(transformation)f(is)h(the)h(Lam)m(b)s(ert)f (equiv)-5 b(alen)m(t)32 b(azim)m(uthal)f(pro)5 b(jection)1365 1667 y Fh(X)91 b Fq(=)83 b(2)17 b(sin)1898 1521 y Fe( )1973 1600 y Fh(\031)p 1973 1644 59 4 v 1978 1736 a Fq(4)2064 1667 y Fo(\000)2174 1600 y Fh(\022)p 2174 1644 49 4 v 2174 1736 a Fq(2)2233 1521 y Fe(!)2315 1667 y Fq(cos)g Fh(\036)1376 1950 y(Y)104 b Fq(=)83 b(2)17 b(sin)1898 1804 y Fe( )1973 1882 y Fh(\031)p 1973 1926 59 4 v 1978 2018 a Fq(4)2064 1950 y Fo(\000)2174 1882 y Fh(\022)p 2174 1926 49 4 v 2174 2018 a Fq(2)2233 1804 y Fe(!)2315 1950 y Fq(sin)g Fh(\036)802 b Fq(\(3.17\))324 2226 y(for)35 b(the)i(North)f(P)m(ole.)54 b(The)37 b(transformation)d(for)i(the)g (South)h(P)m(ole)f(is)f(similar.)51 b(This)324 2347 y(transformation)34 b(is)h(only)h(used)h(to)f(compute)g(in)m(tersections;)i(line)d(in)m (tegrals)g(are)h(still)324 2467 y(computed)28 b(in)f (latitude-longitude)d(co)s(ordinates.)42 b(Because)29 b(in)m(tersections)f(computed)324 2588 y(in)50 b(the)h(transformed)f (co)s(ordinates)g(can)h(b)s(e)f(di\013eren)m(t)h(from)e(those)j (computed)e(in)324 2708 y(latitude-longitude)23 b(co)s(ordinates,)k (line)f(segmen)m(ts)h(whic)m(h)g(cross)h(the)f(latitude)e(thresh-)324 2828 y(old)33 b(m)m(ust)i(b)s(e)f(treated)h(carefully)-8 b(.)48 b(T)-8 b(o)34 b(compute)h(the)f(in)m(tersections)h(consisten)m (tly)g(for)324 2949 y(suc)m(h)41 b(a)f(segmen)m(t,)j(in)m(tersections)e (with)f(the)g(threshold)h(latitude)d(are)j(detected)g(and)324 3069 y(used)33 b(as)g(a)f(normal)f(grid)g(in)m(tersection)h(to)g(pro)m (vide)h(a)f(clean)g(break)i(b)s(et)m(w)m(een)g(the)f(t)m(w)m(o)324 3189 y(co)s(ordinate)e(systems.)324 3522 y Fm(3.5)160 b(Conclusion)324 3741 y Fq(The)27 b(implemen)m(tation)22 b(in)j(the)i(SCRIP)g(co)s(de)f(follo)m(ws)e(closely)i(the)g (description)g(ab)s(o)m(v)m(e.)324 3862 y(The)34 b(user)g(should)f(b)s (e)h(able)f(to)g(follo)m(w)e(and)j(understand)g(the)g(pro)s(cess)g (based)h(on)e(this)324 3982 y(description.)1894 5251 y(20)p eop %%Page: 21 23 21 22 bop 324 1212 a Fn(Chapter)78 b(4)324 1627 y(Bilinear)e(Remapping) 324 2080 y Fq(Standard)44 b(bilinear)d(in)m(terp)s(olation)g(sc)m (hemes)46 b(can)e(b)s(e)g(found)g(in)f(man)m(y)h(textb)s(o)s(oks.)324 2200 y(Here)25 b(w)m(e)g(presen)m(t)g(a)f(more)f(general)h(sc)m(heme)h (whic)m(h)g(uses)g(a)f(lo)s(cal)e(bilinear)f(appro)m(xima-)324 2320 y(tion)k(to)h(in)m(terp)s(olate)f(to)h(a)g(p)s(oin)m(t)f(in)h(a)g (quadrilateral)e(grid.)40 b(Consider)27 b(the)g(grid)e(p)s(oin)m(ts)324 2441 y(sho)m(wn)34 b(in)d(Fig.)h(4.1)g(lab)s(elled)e(with)i (logically-rectangular)c(indices)33 b(\(e.g.)43 b(\()p Fh(i;)17 b(j)6 b Fq(\)\).)470 2561 y(Let)37 b(the)h(latitude-longitude) c(co)s(ordinates)i(of)h(p)s(oin)m(t)f(1)h(b)s(e)g(\()p Fh(\022)s Fq(\()p Fh(i;)17 b(j)6 b Fq(\))p Fh(;)17 b(\036)p Fq(\()p Fh(i;)g(j)6 b Fq(\)\),)37 b(the)324 2682 y(co)s(ordinates)g(of) g(p)s(oin)m(t)f(2)i(b)s(e)f(\()p Fh(\022)s Fq(\()p Fh(i)26 b Fq(+)f(1)p Fh(;)17 b(j)6 b Fq(\))p Fh(;)17 b(\036)p Fq(\()p Fh(i)25 b Fq(+)g(1)p Fh(;)17 b(j)6 b Fq(\)\),)38 b(etc.)59 b(No)m(w)38 b(let)f Fh(\013)h Fq(and)f Fh(\014)43 b Fq(b)s(e)324 2802 y(con)m(tin)m(uous)36 b(lo)s(cal)d(co)s(ordinates)i (suc)m(h)i(that)e(the)h(co)s(ordinates)f(\()p Fh(\013)q(;)17 b(\014)6 b Fq(\))34 b(of)h(p)s(oin)m(t)f(1)i(are)324 2922 y(\(0)p Fh(;)17 b Fq(0\),)34 b(p)s(oin)m(t)f(2)h(are)h(\(1)p Fh(;)17 b Fq(0\),)34 b(p)s(oin)m(t)f(3)h(are)h(\(1)p Fh(;)17 b Fq(1\))33 b(and)i(p)s(oin)m(t)e(4)h(are)g(\(0)p Fh(;)17 b Fq(1\).)48 b(If)35 b(p)s(oin)m(t)e Fh(P)324 3043 y Fq(lies)h(inside)g(the)i(cell)e(formed)h(b)m(y)h(the)f(four)g(p) s(oin)m(ts)g(ab)s(o)m(v)m(e,)i(the)e(function)g Fh(f)46 b Fq(at)34 b(p)s(oin)m(t)324 3163 y Fh(P)46 b Fq(can)33 b(b)s(e)f(appro)m(ximated)g(b)m(y)503 3383 y Fh(f)551 3398 y Ff(P)693 3383 y Fq(=)83 b(\(1)22 b Fo(\000)h Fh(\013)q Fq(\)\(1)e Fo(\000)i Fh(\014)6 b Fq(\))p Fh(f)11 b Fq(\()p Fh(i;)17 b(j)6 b Fq(\))21 b(+)h Fh(\013)q Fq(\(1)f Fo(\000)i Fh(\014)6 b Fq(\))p Fh(f)11 b Fq(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)6 b Fq(\))21 b(+)852 3528 y Fh(\013)q(\014)6 b(f)11 b Fq(\()p Fh(i)21 b Fq(+)h(1)p Fh(;)17 b(j)28 b Fq(+)22 b(1\))g(+)g(\(1)g Fo(\000)g Fh(\013)q Fq(\))p Fh(\014)6 b(f)11 b Fq(\()p Fh(i;)17 b(j)27 b Fq(+)22 b(1\))874 b(\(4.1\))693 3674 y(=)83 b Fh(w)922 3689 y Fi(1)961 3674 y Fh(f)11 b Fq(\()p Fh(i;)17 b(j)6 b Fq(\))22 b(+)g Fh(w)1409 3689 y Fi(2)1448 3674 y Fh(f)11 b Fq(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)6 b Fq(\))22 b(+)g Fh(w)2065 3689 y Fi(3)2104 3674 y Fh(f)11 b Fq(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)27 b Fq(+)c(1\))e(+)h Fh(w)2889 3689 y Fi(4)2929 3674 y Fh(f)11 b Fq(\()p Fh(i;)17 b(j)27 b Fq(+)22 b(1\))p Fh(:)324 3894 y Fq(The)37 b(remapping)f(w)m(eigh)m(ts)h(m)m(ust)g (therefore)g(b)s(e)g(computed)g(b)m(y)h(\014nding)e Fh(\013)h Fq(and)g Fh(\014)42 b Fq(at)324 4014 y(p)s(oin)m(t)31 b Fh(P)14 b Fq(.)470 4134 y(The)38 b(latitude-longitude)33 b(co)s(ordinates)j(\()p Fh(\022)s(;)17 b(\036)p Fq(\))37 b(of)f(p)s(oin)m(t)g Fh(P)50 b Fq(are)36 b(kno)m(wn)i(and)f(can)324 4255 y(also)31 b(b)s(e)i(appro)m(ximated)f(b)m(y)675 4475 y Fh(\022)86 b Fq(=)d(\(1)22 b Fo(\000)h Fh(\013)q Fq(\)\(1)e Fo(\000)i Fh(\014)6 b Fq(\))p Fh(\022)1627 4490 y Fi(1)1688 4475 y Fq(+)22 b Fh(\013)q Fq(\(1)g Fo(\000)g Fh(\014)6 b Fq(\))p Fh(\022)2201 4490 y Fi(2)2263 4475 y Fq(+)22 b Fh(\013)q(\014)6 b(\022)2530 4490 y Fi(3)2591 4475 y Fq(+)22 b(\(1)g Fo(\000)h Fh(\013)q Fq(\))p Fh(\014)6 b(\022)3105 4490 y Fi(4)665 4620 y Fh(\036)83 b Fq(=)g(\(1)22 b Fo(\000)h Fh(\013)q Fq(\)\(1)e Fo(\000)i Fh(\014)6 b Fq(\))p Fh(\036)1640 4635 y Fi(1)1700 4620 y Fq(+)22 b Fh(\013)q Fq(\(1)g Fo(\000)h Fh(\014)6 b Fq(\))p Fh(\036)2227 4635 y Fi(2)2287 4620 y Fq(+)22 b Fh(\013)q(\014)6 b(\036)2567 4635 y Fi(3)2628 4620 y Fq(+)22 b(\(1)g Fo(\000)h Fh(\013)q Fq(\))p Fh(\014)6 b(\036)3155 4635 y Fi(4)3193 4620 y Fh(:)141 b Fq(\(4.2\))324 4840 y(Because)36 b(\(4.2\))e(is)g(nonlinear)f(in)g Fh(\013)j Fq(and)e Fh(\014)6 b Fq(,)35 b(w)m(e)g(m)m(ust)g(linearize)e(and)h (iterate)g(to)m(w)m(ard)1894 5251 y(21)p eop %%Page: 22 24 22 23 bop 1063 1127 a Fq(Figure)31 b(4.1:)43 b(A)33 b(general)f (quadrilateral)e(grid.)656 4143 y Fc(\010)739 4102 y(\010)822 4060 y(\010)905 4019 y(\010)988 3977 y(\010)1071 3936 y(\010)1154 3894 y(\010)1237 3853 y(\010)1320 3811 y(\010)1403 3770 y(\010)1486 3728 y(\010)1569 3687 y(\010)1652 3645 y(\010)1735 3604 y(\010)1818 3562 y(\010)1901 3521 y(\010)1984 3479 y(\010)2067 3438 y(\010)2150 3396 y(\010)2233 3355 y(\010)2316 3313 y(\001)2358 3230 y(\001)2399 3147 y(\001)2441 3064 y(\001)2482 2981 y(\001)2524 2898 y(\001)2565 2815 y(\001)2607 2732 y(\001)2648 2649 y(\001)2690 2566 y(\001)2731 2483 y(\001)2773 2400 y(\001)2814 2317 y(\001)2856 2234 y(\001)2897 2151 y(\001)2939 2068 y(\001)2980 1985 y(\001)3022 1902 y(\001)3064 1819 y(\001)3105 1736 y(\001)656 4143 y(\001)697 4060 y(\001)739 3977 y(\001)780 3894 y(\001)822 3811 y(\001)863 3728 y(\001)905 3645 y(\001)946 3562 y(\001)988 3479 y(\001)1029 3396 y(\001)1071 3313 y(\001)1112 3230 y(\001)1154 3147 y(\001)1196 3064 y(\001)1237 2981 y(\001)1279 2898 y(\001)1320 2815 y(\001)1362 2732 y(\001)1403 2649 y(\001)1445 2566 y(\001)1486 2483 y(\010)1569 2442 y(\010)1652 2400 y(\010)1735 2359 y(\010)1818 2317 y(\010)1901 2275 y(\010)1984 2234 y(\010)2067 2192 y(\010)2150 2151 y(\010)2233 2109 y(\010)2316 2068 y(\010)2399 2026 y(\010)2482 1985 y(\010)2565 1943 y(\010)2648 1902 y(\010)2731 1860 y(\010)2814 1819 y(\010)2897 1777 y(\010)2980 1736 y(\010)3064 1694 y(\010)656 4143 y Fa(s)2316 3313 y(s)1486 2483 y(s)3147 1653 y(s)2067 2815 y(s)573 4310 y Fq(1)i(\()p Fh(i;)17 b(j)6 b Fq(\))2399 3396 y(2)33 b(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)6 b Fq(\))988 2483 y(\()p Fh(i;)17 b(j)28 b Fq(+)22 b(1\))32 b(4)3230 1653 y(3)g(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)28 b Fq(+)22 b(1\))1984 2815 y Fh(P)1894 5251 y Fq(22)p eop %%Page: 23 25 23 24 bop 324 548 a Fq(a)32 b(solution.)42 b(Di\013eren)m(tiating)30 b(\(4.2\))i(results)h(in)1523 684 y Fe(")1618 769 y Fh(\016)t(\022)1613 890 y(\016)t(\036)1759 684 y Fe(#)1836 830 y Fq(=)27 b Fh(A)2029 684 y Fe(")2119 769 y Fh(\016)t(\013)2120 890 y(\016)t(\014)2270 684 y Fe(#)2335 830 y Fh(;)999 b Fq(\(4.3\))324 1107 y(where)324 1364 y Fh(A)28 b Fq(=)528 1218 y Fe(")655 1303 y Fq(\()p Fh(\022)738 1318 y Fi(2)800 1303 y Fo(\000)22 b Fh(\022)944 1318 y Fi(1)984 1303 y Fq(\))g(+)g(\()p Fh(\022)1225 1318 y Fi(1)1287 1303 y Fo(\000)h Fh(\022)1432 1318 y Fi(4)1494 1303 y Fq(+)f Fh(\022)1637 1318 y Fi(3)1699 1303 y Fo(\000)h Fh(\022)1844 1318 y Fi(2)1884 1303 y Fq(\))p Fh(\014)162 b Fq(\()p Fh(\022)2222 1318 y Fi(4)2284 1303 y Fo(\000)22 b Fh(\022)2428 1318 y Fi(1)2468 1303 y Fq(\))g(+)g(\()p Fh(\022)2709 1318 y Fi(1)2771 1303 y Fo(\000)h Fh(\022)2916 1318 y Fi(4)2978 1303 y Fq(+)f Fh(\022)3121 1318 y Fi(3)3183 1303 y Fo(\000)h Fh(\022)3328 1318 y Fi(2)3368 1303 y Fq(\))p Fh(\013)618 1424 y Fq(\()p Fh(\036)714 1439 y Fi(2)775 1424 y Fo(\000)g Fh(\036)933 1439 y Fi(1)972 1424 y Fq(\))f(+)g(\()p Fh(\036)1226 1439 y Fi(1)1287 1424 y Fo(\000)h Fh(\036)1445 1439 y Fi(4)1506 1424 y Fq(+)f Fh(\036)1662 1439 y Fi(3)1724 1424 y Fo(\000)g Fh(\036)1881 1439 y Fi(2)1920 1424 y Fq(\))p Fh(\014)89 b Fq(\()p Fh(\036)2198 1439 y Fi(4)2259 1424 y Fo(\000)23 b Fh(\036)2417 1439 y Fi(1)2456 1424 y Fq(\))f(+)g(\()p Fh(\036)2710 1439 y Fi(1)2771 1424 y Fo(\000)h Fh(\036)2929 1439 y Fi(4)2990 1424 y Fq(+)f Fh(\036)3146 1439 y Fi(3)3208 1424 y Fo(\000)g Fh(\036)3365 1439 y Fi(2)3404 1424 y Fq(\))p Fh(\013)3547 1218 y Fe(#)3612 1364 y Fh(:)3361 1547 y Fq(\(4.4\))324 1667 y(In)m(v)m(erting)33 b(this)f(system,)741 1944 y Fh(\016)t(\013)c Fq(=)982 1795 y Fe(\014)982 1844 y(\014)982 1894 y(\014)982 1944 y(\014)982 1994 y(\014)1056 1883 y Fh(\016)t(\022)128 b Fq(\()p Fh(\022)1359 1898 y Fi(4)1421 1883 y Fo(\000)22 b Fh(\022)1565 1898 y Fi(1)1605 1883 y Fq(\))g(+)g(\()p Fh(\022)1846 1898 y Fi(1)1908 1883 y Fo(\000)h Fh(\022)2053 1898 y Fi(4)2115 1883 y Fq(+)f Fh(\022)2258 1898 y Fi(3)2320 1883 y Fo(\000)g Fh(\022)2464 1898 y Fi(2)2504 1883 y Fq(\))p Fh(\013)1051 2003 y(\016)t(\036)83 b Fq(\()p Fh(\036)1335 2018 y Fi(4)1396 2003 y Fo(\000)23 b Fh(\036)1554 2018 y Fi(1)1593 2003 y Fq(\))f(+)g(\()p Fh(\036)1847 2018 y Fi(1)1908 2003 y Fo(\000)h Fh(\036)2066 2018 y Fi(4)2127 2003 y Fq(+)f Fh(\036)2283 2018 y Fi(3)2344 2003 y Fo(\000)h Fh(\036)2502 2018 y Fi(2)2541 2003 y Fq(\))p Fh(\013)2683 1795 y Fe(\014)2683 1844 y(\014)2683 1894 y(\014)2683 1944 y(\014)2683 1994 y(\014)2733 1944 y Fo(\004)h Fq(det\()p Fh(A)p Fq(\))p Fh(;)217 b Fq(\(4.5\))324 2221 y(and)743 2379 y Fh(\016)t(\014)33 b Fq(=)982 2229 y Fe(\014)982 2279 y(\014)982 2329 y(\014)982 2379 y(\014)982 2428 y(\014)1088 2318 y Fq(\()p Fh(\022)1171 2333 y Fi(2)1233 2318 y Fo(\000)23 b Fh(\022)1378 2333 y Fi(1)1417 2318 y Fq(\))f(+)g(\()p Fh(\022)1658 2333 y Fi(1)1720 2318 y Fo(\000)h Fh(\022)1865 2333 y Fi(4)1927 2318 y Fq(+)f Fh(\022)2070 2333 y Fi(3)2132 2318 y Fo(\000)h Fh(\022)2277 2333 y Fi(2)2317 2318 y Fq(\))p Fh(\014)130 b(\016)t(\022)1051 2438 y Fq(\()p Fh(\036)1147 2453 y Fi(2)1208 2438 y Fo(\000)23 b Fh(\036)1366 2453 y Fi(1)1405 2438 y Fq(\))f(+)g(\()p Fh(\036)1659 2453 y Fi(1)1720 2438 y Fo(\000)h Fh(\036)1878 2453 y Fi(4)1939 2438 y Fq(+)f Fh(\036)2095 2453 y Fi(3)2157 2438 y Fo(\000)g Fh(\036)2314 2453 y Fi(2)2354 2438 y Fq(\))p Fh(\014)88 b(\016)t(\036)2681 2229 y Fe(\014)2681 2279 y(\014)2681 2329 y(\014)2681 2379 y(\014)2681 2428 y(\014)2731 2379 y Fo(\004)24 b Fq(det\()p Fh(A)p Fq(\))p Fh(:)219 b Fq(\(4.6\))324 2615 y(Starting)34 b(with)i(an)f(initial)d(guess)37 b(for)e Fh(\013)i Fq(and)f Fh(\014)41 b Fq(\(sa)m(y)c Fh(\013)c Fq(=)g Fh(\014)39 b Fq(=)33 b(0\),)j(equations)g(\(4.5\))324 2736 y(and)j(\(4.6\))f(can)h(b)s(e)h(iterated)e(un)m(til)g Fh(\016)t(\013)h Fq(and)g Fh(\016)t(\014)45 b Fq(are)39 b(suitably)f(small.)60 b(The)40 b(w)m(eigh)m(ts)324 2856 y(can)g(then)i(b)s(e)e(computed)h(from)e(\(4.1\).)66 b(Note)41 b(that)f(for)g(simple)f(latitude-longitude)324 2976 y(grids,)32 b(this)g(iteration)f(will)f(con)m(v)m(erge)k(in)e(the) h(\014rst)g(iteration.)470 3097 y(In)39 b(order)f(to)g(compute)g(the)h (w)m(eigh)m(ts)g(using)f(this)f(general)h(bilinear)e(iteration,)i(it) 324 3217 y(m)m(ust)f(b)s(e)g(determined)g(in)f(whic)m(h)h(b)s(o)m(x)g (the)h(p)s(oin)m(t)e Fh(P)50 b Fq(resides.)57 b(F)-8 b(or)36 b(this,)i(the)f(searc)m(h)324 3337 y(algorithms)27 b(outlined)h(in)g(the)i(previous)g(c)m(hapter)h(are)e(used)i(with)e (the)h(exception)g(that)324 3458 y(instead)46 b(of)g(using)h(cell)e (corners,)51 b(the)c(relev)-5 b(an)m(t)46 b(b)s(o)m(x)h(is)f(formed)g (b)m(y)i(neigh)m(b)s(or)e(cell)324 3578 y(cen)m(ters)34 b(as)f(sho)m(wn)h(in)e(Fig.)f(4.1.)1894 5251 y(23)p eop %%Page: 24 26 24 25 bop 324 1212 a Fn(Chapter)78 b(5)324 1627 y(Bicubic)f(Remapping) 324 2080 y Fq(The)42 b(bicubic)e(remapping)g(exactly)h(follo)m(ws)f (the)h(bilinear)e(remapping)g(except)k(that)324 2200 y(four)29 b(w)m(eigh)m(ts)i(for)e(eac)m(h)i(corner)g(p)s(oin)m(t)e(are) g(required.)44 b(Th)m(us,)32 b(n)m(um)p 2824 2200 30 4 v 35 w(wts)e(is)g(set)g(to)g(four)324 2320 y(for)i(this)g(option.)42 b(The)34 b(bicubic)e(remapping)f(is)875 2540 y Fh(f)923 2555 y Ff(P)1065 2540 y Fq(=)83 b(\(1)22 b Fo(\000)g Fh(\014)1493 2499 y Fi(2)1532 2540 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\014)6 b Fq(\)\)\(1)21 b Fo(\000)i Fh(\013)2198 2499 y Fi(2)2237 2540 y Fq(\(3)f Fo(\000)g Fq(2)p Fh(\013)q Fq(\)\))p Fh(f)11 b Fq(\()p Fh(i;)17 b(j)6 b Fq(\))21 b(+)1224 2686 y(\(1)h Fo(\000)g Fh(\014)1493 2645 y Fi(2)1532 2686 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\014)6 b Fq(\)\))p Fh(\013)1990 2645 y Fi(2)2028 2686 y Fq(\(3)22 b Fo(\000)h Fq(2)p Fh(\013)q Fq(\))p Fh(f)11 b Fq(\()p Fh(i)21 b Fq(+)h(1)p Fh(;)17 b(j)6 b Fq(\))22 b(+)1224 2831 y Fh(\014)1285 2790 y Fi(2)1324 2831 y Fq(\(3)g Fo(\000)g Fq(2)p Fh(\014)6 b Fq(\))p Fh(\013)1743 2790 y Fi(2)1782 2831 y Fq(\(3)22 b Fo(\000)g Fq(2)p Fh(\013)q Fq(\))p Fh(f)11 b Fq(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)27 b Fq(+)22 b(1\))g(+)1224 2976 y Fh(\014)1285 2935 y Fi(2)1324 2976 y Fq(\(3)g Fo(\000)g Fq(2)p Fh(\014)6 b Fq(\)\(1)22 b Fo(\000)g Fh(\013)1951 2935 y Fi(2)1990 2976 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\013)q Fq(\)\))p Fh(f)11 b Fq(\()p Fh(i;)17 b(j)27 b Fq(+)22 b(1\))g(+)1224 3171 y(\(1)g Fo(\000)g Fh(\014)1493 3130 y Fi(2)1532 3171 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\014)6 b Fq(\)\))p Fh(\013)q Fq(\()p Fh(\013)22 b Fo(\000)g Fq(1\))2298 3130 y Fi(2)2347 3104 y Fh(@)5 b(f)p 2347 3148 116 4 v 2360 3239 a(@)g(i)2473 3171 y Fq(\()p Fh(i;)17 b(j)6 b Fq(\))22 b(+)1224 3409 y(\(1)g Fo(\000)g Fh(\014)1493 3368 y Fi(2)1532 3409 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\014)6 b Fq(\)\))p Fh(\013)1990 3368 y Fi(2)2028 3409 y Fq(\()p Fh(\013)23 b Fo(\000)g Fq(1\))2348 3342 y Fh(@)5 b(f)p 2348 3386 V 2361 3478 a(@)g(i)2473 3409 y Fq(\()p Fh(i)22 b Fq(+)g(1)p Fh(;)17 b(j)6 b Fq(\))22 b(+)1224 3647 y Fh(\014)1285 3606 y Fi(2)1324 3647 y Fq(\(3)g Fo(\000)g Fq(2)p Fh(\014)6 b Fq(\))p Fh(\013)1743 3606 y Fi(2)1782 3647 y Fq(\()p Fh(\013)23 b Fo(\000)f Fq(1\))2101 3580 y Fh(@)5 b(f)p 2101 3624 V 2114 3716 a(@)g(i)2226 3647 y Fq(\()p Fh(i)23 b Fq(+)f(1)p Fh(;)17 b(j)27 b Fq(+)22 b(1\))g(+)1224 3885 y Fh(\014)1285 3844 y Fi(2)1324 3885 y Fq(\(3)g Fo(\000)g Fq(2)p Fh(\014)6 b Fq(\))p Fh(\013)q Fq(\()p Fh(\013)22 b Fo(\000)h Fq(1\))2052 3844 y Fi(2)2101 3818 y Fh(@)5 b(f)p 2101 3862 V 2114 3954 a(@)g(i)2226 3885 y Fq(\()p Fh(i;)17 b(j)29 b Fq(+)22 b(1\))f(+)1224 4124 y Fh(\014)6 b Fq(\()p Fh(\014)27 b Fo(\000)c Fq(1\))1592 4082 y Fi(2)1631 4124 y Fq(\(1)f Fo(\000)g Fh(\013)1902 4082 y Fi(2)1942 4124 y Fq(\(3)f Fo(\000)i Fq(2)p Fh(\013)q Fq(\)\))2348 4056 y Fh(@)5 b(f)p 2348 4100 V 2354 4192 a(@)g(j)2473 4124 y Fq(\()p Fh(i;)17 b(j)6 b Fq(\))22 b(+)1224 4381 y Fh(\014)6 b Fq(\()p Fh(\014)27 b Fo(\000)c Fq(1\))1592 4340 y Fi(2)1631 4381 y Fh(\013)1694 4340 y Fi(2)1733 4381 y Fq(\(3)f Fo(\000)h Fq(2)p Fh(\013)q Fq(\))2102 4314 y Fh(@)5 b(f)p 2102 4358 V 2108 4449 a(@)g(j)2226 4381 y Fq(\()p Fh(i)23 b Fq(+)f(1)p Fh(;)17 b(j)6 b Fq(\))21 b(+)1224 4639 y Fh(\014)1285 4597 y Fi(2)1324 4639 y Fq(\()p Fh(\014)28 b Fo(\000)22 b Fq(1\))p Fh(\013)1694 4597 y Fi(2)1733 4639 y Fq(\(3)g Fo(\000)h Fq(2)p Fh(\013)q Fq(\))2102 4571 y Fh(@)5 b(f)p 2102 4615 V 2108 4707 a(@)g(j)2226 4639 y Fq(\()p Fh(i)23 b Fq(+)f(1)p Fh(;)17 b(j)27 b Fq(+)22 b(1\))g(+)1224 4896 y Fh(\014)1285 4855 y Fi(2)1324 4896 y Fq(\()p Fh(\014)28 b Fo(\000)22 b Fq(1\)\(1)g Fo(\000)g Fh(\013)1902 4855 y Fi(2)1942 4896 y Fq(\(3)f Fo(\000)i Fq(2)p Fh(\013)q Fq(\)\))2348 4829 y Fh(@)5 b(f)p 2348 4873 V 2354 4964 a(@)g(j)2473 4896 y Fq(\()p Fh(i;)17 b(j)28 b Fq(+)22 b(1\))g(+)1894 5251 y(24)p eop %%Page: 25 27 25 26 bop 1224 595 a Fh(\013)q Fq(\()p Fh(\013)22 b Fo(\000)h Fq(1\))1596 554 y Fi(2)1635 595 y Fh(\014)6 b Fq(\()p Fh(\014)27 b Fo(\000)c Fq(1\))2003 554 y Fi(2)2071 527 y Fh(@)2127 491 y Fi(2)2167 527 y Fh(f)p 2052 571 194 4 v 2052 663 a(@)5 b(i@)g(j)2255 595 y Fq(\()p Fh(i;)17 b(j)6 b Fq(\))22 b(+)1224 862 y Fh(\013)1287 821 y Fi(2)1326 862 y Fq(\()p Fh(\013)h Fo(\000)f Fq(1\))p Fh(\014)6 b Fq(\()p Fh(\014)27 b Fo(\000)c Fq(1\))2003 821 y Fi(2)2071 795 y Fh(@)2127 758 y Fi(2)2167 795 y Fh(f)p 2052 839 V 2052 930 a(@)5 b(i@)g(j)2255 862 y Fq(\()p Fh(i)23 b Fq(+)f(1)p Fh(;)17 b(j)6 b Fq(\))21 b(+)1224 1129 y Fh(\013)1287 1088 y Fi(2)1326 1129 y Fq(\()p Fh(\013)i Fo(\000)f Fq(1\))p Fh(\014)1696 1088 y Fi(2)1735 1129 y Fq(\()p Fh(\014)28 b Fo(\000)22 b Fq(1\))2071 1062 y Fh(@)2127 1026 y Fi(2)2167 1062 y Fh(f)p 2052 1106 V 2052 1198 a(@)5 b(i@)g(j)2255 1129 y Fq(\()p Fh(i)23 b Fq(+)f(1)p Fh(;)17 b(j)27 b Fq(+)22 b(1\))g(+)1224 1396 y Fh(\013)q Fq(\()p Fh(\013)g Fo(\000)h Fq(1\))1596 1355 y Fi(2)1635 1396 y Fh(\014)1696 1355 y Fi(2)1735 1396 y Fq(\()p Fh(\014)28 b Fo(\000)22 b Fq(1\))2071 1329 y Fh(@)2127 1293 y Fi(2)2167 1329 y Fh(f)p 2052 1373 V 2052 1465 a(@)5 b(i@)g(j)2255 1396 y Fq(\()p Fh(i;)17 b(j)28 b Fq(+)22 b(1\))738 b(\(5.1\))324 1661 y(where)30 b Fh(\013)f Fq(and)g Fh(\014)34 b Fq(are)28 b(iden)m(tical)f(to)i (those)g(found)f(in)g(the)h(bilinear)e(case)i(and)g(are)f(found)324 1782 y(using)k(an)h(iden)m(tical)e(algorithm.)42 b(Note)33 b(that)f(unlik)m(e)h(the)g(conserv)-5 b(ativ)m(e)34 b(remappings,)324 1902 y(the)h(gradien)m(ts)f(here)i(are)e(gradien)m(ts)h(with)f(resp)s (ect)i(to)e(the)h Fg(lo)-5 b(gic)g(al)33 b Fq(v)-5 b(ariable)33 b(and)i(not)324 2022 y(latitude)28 b(or)h(longitude.)42 b(Lastly)-8 b(,)29 b(the)i(four)e(w)m(eigh)m(ts)i(corresp)s(onding)e (to)h(eac)m(h)g(address)324 2143 y(pair)39 b(corresp)s(ond)h(to)g(the)g (w)m(eigh)m(t)g(m)m(ultiplying)d(the)j(\014eld)f(v)-5 b(alue)40 b(at)f(the)h(p)s(oin)m(t,)h(the)324 2263 y(w)m(eigh)m(t)29 b(m)m(ultiplying)c(the)k(gradien)m(t)f(with)h(resp)s(ect)h(to)e Fh(i)p Fq(,)i(the)f(w)m(eigh)m(t)g(m)m(ultiplying)c(the)324 2384 y(gradien)m(t)33 b(with)h(resp)s(ect)h(to)f Fh(j)6 b Fq(,)34 b(and)g(the)h(w)m(eigh)m(t)f(m)m(ultiplying)d(the)j(cross)h (gradien)m(t)e(in)324 2504 y(that)f(order.)1894 5251 y(25)p eop %%Page: 26 28 26 27 bop 324 1212 a Fn(Chapter)78 b(6)324 1627 y(Distance-w)-6 b(eigh)g(ted)77 b(Av)-6 b(erage)324 1876 y(Remapping)324 2329 y Fq(This)42 b(sc)m(heme)h(for)e(remapping)g(is)g(probably)h(the)g (simplest)f(in)g(this)h(pac)m(k)-5 b(age.)72 b(The)324 2449 y(co)s(de)30 b(simply)f(searc)m(hes)j(for)d(the)i(n)m(um)p 1736 2449 30 4 v 35 w(neigh)m(b)s(ors)f(nearest)h(neigh)m(b)s(ors)f (and)g(computes)324 2570 y(the)j(w)m(eigh)m(ts)g(using)1297 2739 y Fh(w)d Fq(=)1839 2672 y(1)p Fh(=)p Fq(\()p Fh(d)21 b Fq(+)h Fh(\017)p Fq(\))p 1510 2716 1042 4 v 1510 2746 a Fe(P)1598 2772 y Fi(n)n(um)p 1740 2772 30 4 v 34 w(neigh)n(b)r(ors) 1598 2837 y Ff(n)2067 2812 y Fq([1)p Fh(=)p Fq(\()p Fh(d)2281 2827 y Ff(n)2349 2812 y Fq(+)g Fh(\017)p Fq(\)])2562 2739 y Fh(;)772 b Fq(\(6.1\))324 2969 y(where)29 b Fh(\017)f Fq(is)f(a)h(small)d(n)m(um)m(b)s(er)j(to)f(prev)m(en)m(t)j(dividing)c (b)m(y)j(zero,)g(the)f(sum)g(is)f(for)g(normal-)324 3089 y(ization)37 b(and)j Fh(d)f Fq(is)g(the)g(distance)h(from)e(the)i (destination)e(grid)g(p)s(oin)m(t)h(to)g(the)h(source)324 3209 y(grid)31 b(p)s(oin)m(t.)43 b(The)33 b(distance)g(is)f(the)h (angular)f(distance)452 3429 y Fh(d)27 b Fq(=)h(cos)764 3388 y Fj(\000)p Fi(1)875 3429 y Fq(\(cos)17 b Fh(\022)1105 3444 y Ff(d)1163 3429 y Fq(cos)g Fh(\022)1355 3444 y Ff(s)1392 3429 y Fq(\(cos)g Fh(\036)1635 3444 y Ff(d)1692 3429 y Fq(cos)g Fh(\036)1897 3444 y Ff(s)1956 3429 y Fq(+)22 b(sin)16 b Fh(\036)2248 3444 y Ff(d)2305 3429 y Fq(sin)h Fh(\036)2500 3444 y Ff(s)2536 3429 y Fq(\))22 b(+)g(sin)17 b Fh(\022)2876 3444 y Ff(d)2933 3429 y Fq(sin)f Fh(\022)3114 3444 y Ff(s)3152 3429 y Fq(\))g Fh(;)128 b Fq(\(6.2\))324 3649 y(where)33 b Fh(\022)j Fq(is)c(latitude,)f Fh(\036)h Fq(is)f(longitude)g(and)i(the)f(subscripts)i Fh(d;)17 b(s)32 b Fq(denote)h(destination)324 3770 y(and)f(source)i (grids,)e(resp)s(ectiv)m(ely)-8 b(.)470 3890 y(When)41 b(\014nding)e(nearest)i(neigh)m(b)s(ors,)g(the)f(distance)g(is)f(not)h (computed)g(b)s(et)m(w)m(een)324 4011 y(the)30 b(destination)f(grid)g (p)s(oin)m(t)g(and)h(ev)m(ery)h(source)g(grid)e(p)s(oin)m(t.)42 b(Instead,)32 b(the)e(searc)m(h)h(is)324 4131 y(narro)m(w)m(ed)i(b)m(y) g(sorting)e(the)i(t)m(w)m(o)f(grids)g(in)m(to)f(latitude)g(bins.)43 b(Only)32 b(those)h(source)g(grid)324 4251 y(cells)28 b(lying)g(in)h(the)h(same)f(latitude)f(bin)g(as)i(the)g(destination)e (grid)g(cell)g(are)i(c)m(hec)m(k)m(ed)i(to)324 4372 y(\014nd)h(the)g (nearest)g(neigh)m(b)s(ors.)1894 5251 y(26)p eop %%Page: 27 29 27 28 bop 324 1212 a Fn(Chapter)78 b(7)324 1627 y(Bugs)324 2080 y Fq(A)43 b(\014le)f(called)f(`bugs')j(is)e(included)g(in)g(the)i (distribution)c(whic)m(h)k(lists)d(curren)m(t)j(out-)324 2200 y(standing)35 b(bugs)h(as)f(w)m(ell)g(as)h(a)f(v)m(ersion)h (history)-8 b(.)52 b(An)m(y)36 b(further)g(bugs,)h(commen)m(ts,)f(or) 324 2320 y(suggestions)d(should)f(b)s(e)h(sen)m(t)h(to)e(me)g(at)g(p)m (wjones@lanl.go)m(v.)470 2441 y(The)d(co)s(de)g(do)s(es)f(not)h(ha)m(v) m(e)g(v)m(ery)h(useful)e(error)g(messages)h(to)f(help)g(diagnose)g (prob-)324 2561 y(lems)k(so)g(feel)g(free)h(to)g(p)s(ester)g(me)f(with) g(an)m(y)i(problems)e(y)m(ou)h(encoun)m(ter.)470 2682 y(The)25 b(pac)m(k)-5 b(age)25 b(has)g(also)f(not)g(b)s(een)h(extensiv) m(ely)h(tested)g(on)e(a)g(v)-5 b(ariet)m(y)25 b(of)f(mac)m(hines.)324 2802 y(It)33 b(w)m(orks)i(\014ne)g(on)e(SGI)h(mac)m(hines)f(and)h(IBM)g (mac)m(hines,)g(but)g(has)g(not)f(b)s(een)i(run)f(on)324 2922 y(other)41 b(mac)m(hines.)68 b(It)41 b(is)f(prett)m(y)i(v)-5 b(anilla)38 b(F)-8 b(ortran,)42 b(so)f(should)g(b)s(e)g(ok)g(on)g(mac)m (hines)324 3043 y(with)32 b(standard-complian)m(t)f(F90)g(compilers.) 1894 5251 y(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF